Question

Let \(X\) be a random sample of size \(1\) from a \(U(0,\theta)\) distribution, where \(\theta>0\) is an unknown parameter. To test \(H_0:\theta\geq 1\) against \(H_1:\theta<1\), the critical region \(X<\frac{3}{4}\) is being used. If \(\beta(\cdot)\) is the power function of the test, then which one of the following statements is NOT true?

• A. The size of the test is \(\dfrac{3}{4}\)
• B. \(\inf_{\theta<1}\beta(\theta)=\dfrac{3}{4}\)
• C. For all \(\theta>0,\;\beta(\theta)<1\)
• D. \(\displaystyle \lim_{\theta\to 0}\beta(\theta)=1\)

Hint:

For a \(U(0,\theta)\) distribution, probabilities are obtained by dividing interval length by \(\theta\). Always write the power function piecewise depending on the relation between the cutoff and \(\theta\).

Answer 1

The Correct Option is C

Step 1: Find the power function.
Since
\[ X\sim U(0,\theta), \] the density function is
\[ f(x)=\frac{1}{\theta}, \qquad 0<x<\theta \] The critical region is
\[ X<\frac34 \] Hence, the power function is
\[ \beta(\theta) = P_\theta\left(X<\frac34\right) \]
For a uniform distribution,
\[ P_\theta(X<a)= \begin{cases} \dfrac{a}{\theta}, & a<\theta,\\ 1, & a\geq \theta. \end{cases} \] Taking
\[ a=\frac34, \] we get
\[ \beta(\theta)= \begin{cases} 1, & 0<\theta\leq \dfrac34,\\ \dfrac{3}{4\theta}, & \theta>\dfrac34. \end{cases} \]

Step 2: Check option (A).
The size of the test is
\[ \sup_{\theta\geq 1}\beta(\theta) \] For
\[ \theta\geq 1, \] we use
\[ \beta(\theta)=\frac{3}{4\theta} \] This function decreases as \(\theta\) increases. Hence, the supremum occurs at
\[ \theta=1 \] Therefore,
\[ \text{Size}=\beta(1)=\frac34 \] So, option (A) is true.

Step 3: Check option (B).
For
\[ \theta<1, \] the power function is
\[ \beta(\theta)= \begin{cases} 1, & 0<\theta\leq \dfrac34,\\ \dfrac{3}{4\theta}, & \dfrac34<\theta<1. \end{cases} \] Now,
\[ \frac{3}{4\theta} \] decreases as \(\theta\to 1^{-}\).
Thus,
\[ \inf_{\theta<1}\beta(\theta) = \lim_{\theta\to 1^-}\frac{3}{4\theta} = \frac34 \] Hence, option (B) is true.

Step 4: Check option (C).
If
\[ 0<\theta\leq \frac34, \] then
\[ \beta(\theta)=1 \] Thus, it is not true that
\[ \beta(\theta)<1 \] for all \(\theta>0\).
Hence, option (C) is NOT true.

Step 5: Check option (D).
As
\[ \theta\to 0, \] eventually
\[ \theta<\frac34 \] and therefore
\[ \beta(\theta)=1 \] Hence,
\[ \lim_{\theta\to 0}\beta(\theta)=1 \] So, option (D) is true.

Step 6: Final conclusion.
The statement which is NOT true is
\[ \boxed{(C)} \]